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Contrast, constancy and measurements of perceived

lightness under parametric manipulation of surface

slant and surface reflectance

Sarah R. Allred,1,! and David H. Brainard,1

1Department of Psychology, University of Pennsylvania,

3401 Walnut Street, 302C, Philadelphia, PA 19104

!Corresponding author: [email protected]

For perceived surface lightness to be a useful guide to object identity, it should

correlate with surface reflectance. Across illumination changes, local contrast

provides a valid cue to surface reflectance. However, when the surfaces sur-

rounding an object of interest change, local contrast is not necessarily a valid

cue. To generalize beyond theories that rely on local contrast as the sole ex-

planatory construct, we require an empirical account of which stimulus factors

produce e!ects beyond those explainable in terms of local contrast. A fruit-

ful approach is to hold local contrast fixed while varying other aspects of

the stimulus. Here we adopted this approach to study e!ects of object slant

in three-dimensional scenes. Observers viewed real, illuminated objects and

matched the apparent lightness of a variable match spot on one surface to a

fixed reference spot on another surface. We parametrically varied the surface

slant and local surround reflectance of the match spot. When local contrast

was a valid cue to test spot reflectance, all observers were approximately light-

ness constant. When local contrast was not a valid cue to test spot reflectance,

observers’ lightness matches were intermediate between contrast matching and

lightness constancy. Quantitative model comparison showed that for most ob-

servers, surface slant exerted an e!ect on perceived lightness beyond that ex-

plainable by the photometric properties of the local surround.

c" 2008 Optical Society of America

OCIS codes: 330.5020, 330.5510

2

1. Introduction

For perceived surface color to be a useful guide to object identity, it should correlate with

surface reflectance. This is di"cult to achieve because the sensory signal that reaches the

eye confounds surface reflectance with the illuminant. For example, the light reflected from a

ripe banana under bright mid-day sunlight is very di!erent than under cloudy, late afternoon

sunlight. The ability of the visual system to maintain a stable perception of surface color

across changes in viewing conditions is called color constancy.

A large body of literature confirms that human vision exhibits approximate color constancy

across changes in illumination (e.g. [1], [2], [3], [4], [5], [6]). A feature of most experiments

is that a test object is viewed in the context of a broader scene, and the illuminant is

manipulated while the objects surrounding the test are held fixed. Under these conditions,

approximate color constancy may be achieved if the brain codes color through some sort of

ratio between the light reflected from the test and that reflected from nearby objects ( [7], [8],

[9]). For example, the ratio of cone responses to the light reflected from neighboring surfaces

is approximately invariant with respect to changes in illumination ( [10], [11], [12], [13]).

The notion that perceived color and lightness at an image location depends on a ratio-like

comparison between the stimulus at that location and at neighboring locations is at the core

of many theories of color and lightness perception ( [7], [14], [15], [16], [17], [18]). We will

refer to this general idea as local contrast coding. Local contrast coding provides an intuitive

explanation for some illusions (e.g. simultaneous contrast, see [19], [20] for discussion). It is

also invoked to explain data from single-unit electrophysiology in the retina, the LGN and

primary visual cortex (see [21]).

If all that ever changed in a scene were the illuminant, then local contrast would always

provide a valid cue to object surface reflectance. Indeed, if the surfaces in the viewed scene

never vary, achieving constancy would not be a challenging computational problem. What

makes constancy di"cult is that fact that both the illuminant and the contextual surfaces

in the scene can change. For example, bananas fall from the plant to the ground. When the

objects surrounding an object of interest change, local contrast is not necessarily a valid cue

to surface reflectance. And the fact that local contrast does not always predict appearance

is evident in various visual illusions (e.g. Adelson’s checkerboard illusion, White’s Illusion,

again see [19], [20]). Experimental studies that explicitly separate e!ects of changing the

illuminant from those of changing the local surround also show that knowing contrast alone

is not su"cient to predict perceived color and lightness ( [22], [23], [24]).

Although it is clear that we need to generalize beyond theories that rely solely on local

contrast as the explanatory construct, the empirical foundations for such generalization re-

main to be established. An important agenda is to understand what stimulus factors produce

e!ects beyond those explainable in terms of local contrast. To this end, a fruitful approach is

3

Fig. 1. Observer’s view of experimental setup. The circular stages on which

the cards rested could be rotated.

to hold local contrast fixed while varying other aspects of the stimulus (e.g. [23], [24], [25]).

Here we adopt this approach to study e!ects of object pose in three-dimensional scenes.

Hochberg and Beck ( [26], see also [27], [28], [29]) showed that manipulations which changed

the perceived pose of a surface relative to a directional light source, while holding the stimulus

constant, also changed its perceived lightness. This allowed them to demonstrate that the

lightness e!ect was driven by the perceived scene layout, with local contrast held constant.

More recent work has studied this type of e!ect parametrically ( [30], [31], [32]), providing

data that allow development and evaluation of quantitative models (e.g. [30], [32], [33]).

This parametric work does not, however, separate e!ects of local contrast from those of

geometry. To clarify the interaction of these two factors, we report experiments that combine

manipulation of surface pose and of local contrast.

2. Methods

2.A. Observers

Observers were six adults between 20 and 35. Observers FP, HB and IY were paid volunteers

who were naive to the purposes of the experiment and had little experience in psychophysical

observations. The other observers (SRA, DBH, RTO) were lab members with varying degrees

of familiarity with experimental design and aims. Note that observer DBH is not the second

author (DHB).

4

2.B. Apparatus

Observers looked through an aperture into an experimental chamber to view two stages,

as shown in Figure 1. Observers were seated 1.3 m from the stages. Ambient illumination

was provided via a single incandescent theater bulb mounted above and to the left of the

observer. Light from the bulb passed through a blue filter, and the voltage to the bulb was

computer-controlled. In addition, illumination to part of the booth was manipulated via a

hidden projector (EPSON PowerLite 8200i), which was also computer-controlled. At RGB

settings of [0,0,0], the projector cast some light, and this was included in our calculations

of the ambient illuminant. With the projector at [0, 0, 0] and the incandescent light at its

normal experimental settings, a white piece of paper on the left stage reflected light of CIE

xyY coordinates of (0.41, 0.41, 402 cd/m2); the same paper on the right stage reflected light

with CIE xyY coordinates of (0.41, 0.41, 261 cd/m2). A box covered with black felt was

placed in the booth and mounted on four adjustable silver feet, two of which can be seen

in Figure 1. A small rectangular slit was cut in the box, and the box was carefully adjusted

until the edge of the light generated by the hidden projector vanished through the thin slit.

This light trap served to minimize observers’ awareness of the hidden projector.

2.C. Stimuli

Stimuli were constructed by printing a standard gray surface (reflectance = 0.12) of size 6 cm

by 6 cm centered on a Mondrian pattern (18 reflectance values, range: 0.02 to 1). Identical

Mondrians (see Figure 1) were mounted on two rotatable stages; a reference stage on the

left and a match stage on the right. At standard viewing distance (1.3 meters) the central

surfaces subtended 2.6 " of visual angle.

Background surfaces of di!erent reflectance were simulated by using the hidden projector

in the following fashion.

2.C.1. Simulating surfaces

First, we created six surfaces (reflectances: 0.12, 0.20, 0.31, 0.57, 0.72, 1). With the exper-

imental lights on and the projector at its miminum settings [0, 0, 0], we took radiometer

readings (PR650) of each surface at the reference location at 0 " slant. We then took radiome-

ter readings of each surface at the match location at each of five di!erent surface slants (0 ",

5 ", 10 ", 15 ", 20 "). The measured chromaticity did not change with location or surface slant.

We then fit the measured CIE xyY values as a function of the reflectance of the surface un-

der question. In each case the data were well-fit by a second order polynomial. From this

function, we could calculate the predicted CIE xyY values for a surface of any reflectance.

We repeated this procedure for each of the match slants. From these fits, we could predict

the CIE xyY values at each surface slant for a surface of any reflectance.

5

We then simulated surfaces by combining a standard surface with projector values cali-

brated to produce the expected CIE xyY values of the real surfaces. To do this, we created a

standard surface (reflectance = 0.12) and took radiometer measurements of that surface at

each slant under di!erent projector settings. We repeated the measurements three times and

used the average. Using algorithms from the Psychophysics Toolbox [34], we fit measured

values to projector settings with cubic splines, and we used these to map between projector

values and CIE xyY values.

The range and precision of surfaces that could be simulated were limited by the gamut

and resolution of the projector. Because the standard surface had reflectance (0.12) we

could not in principle simulate surfaces less reflective than that. Empirically, we were able to

simulate adequately surfaces of reflectance 0.13 to 0.94. Because the intensity of the projector

changes in discrete steps, rather than continuously, the precision with which surfaces could be

simulated was limited. Because of this, we report measured CIE xyY values and reflectance

values actually obtained rather than requested values. Although we could not simulate every

reflectance perfectly, we could simulate changes in reflectance of about half a percent of the

requested reflectance.

On each day, a geometry file determining the location of the projected light was re-

calibrated. The small black felt strip around the inner edge of the Mondrian (Figure 1)

served to hide any residual misalignment.

In the experiments, spots of di!erent reflectance were simulated by projecting onto the

the reference and match squares. At 0 " slant, spots were circles with a radius of 1.25 cm and

at standard viewing distance they subtended 1.1 " of visual angle. As with the geometry of

the projected squares, the geometry of the projected circles was manipulated with changing

slant to simulate a physically rotating circle.

Hereafter, we refer to manipulations of match and reference simply as reflectance changes

rather than as simulated reflectance changes. All reported luminance values were measured

in situ.

2.D. Psychophysical Task

Observers initiated a block of trials by pressing a key, after which a shutter opened to

reveal the experimental booth (Figure 1). On each trial, observers performed the following

2AFC psychophysical task. Two spots were presented, one at the center of the reference

surround (gray square on the left stage, Figure 1) and one at the center of the match

surround (gray square on the right stage, Figure 1). Observers were instructed to move a

joystick to indicate which spot appeared lighter. Reference and match spots were presented

for 1500 ms, accompanied by a 250 Hz tone. Across all blocks of trials, the reference surround

reflectance was fixed at 0.16 and the reference stage was fixed at 0 " slant. Match surround

6

reflectance and match slant were varied between blocks of trials, but remained fixed within

a block of trials. Match surround reflectance took values of 0.16, 0.25, 0.34, 0.44, 0.56,

and match slant took values of 0 ", 10 " and 20 ". Match surround reflectance and slant

were parametrically varied, yielding 15 possible match conditions. Within each block of

trials (one match surround reflectance / slant condition), we calculated a point of subjective

equality (PSE) for 5 di!erent reference spots (reflectance values = 0.18, 0.20, 0.22, 0.26,

0.32). One match surround condition (reflectance = 0.34) was added midway through the

experiment, and two subjects (IY, HB) were not tested in this condition. All reference spots

were increments.

The procedure for each reference spot was as follows: On each trial, the reflectance of

the match spot was determined by implementing an adaptive staircase calculated by the

QUEST algorithm ( [35]) as implemented in the Psychophysics Toolbox ( [34]). For each

of the 5 reference spots, we ran three interleaved staircases (10 trials each) with di!erent

target response probabilities (25%, 50%, 75%). In each experimental session, observers ran

between four and seven blocks of 150 trials each. A block lasted approximately six minutes,

and included trials for one match slant and one match surround reflectance.

Between blocks, the shutter closed while the experimenter initiated a new block of trials

with a di!erent match slant and match surround reflectance. Observers were o!ered the

opportunity to take a break in between each block of trials, and each experimental session

lasted between 35 minutes and 1 hour.

Before any experimental data were collected, each observer underwent an induction proce-

dure designed to encourage a strategy of matching surface reflectance rather than luminance

or contrast. In a separate experimental room, observers were seated in front of a plywood

box that had been divided in two, with each side illuminated by a single directional light

source. The right side of the box contained a paint palette, and the left side of the box

contained three cubes, each of which were painted a di!erent shade of gray. While looking

at the cubes, observers were told that in the experiment, they would be matching painted

surfaces or simulations of such surfaces. Observers were instructed to hold the painted cubes

and view them in di!erent orientations and locations within the plywood box. Subsequently,

observers were shown fixed cubes with only one painted surface, and asked to pick the same

paint from the palette.

2.E. Data Analysis and Predictions

Within a block of trials, we fit the probability of reporting that the match spot was lighter

as a function of match spot reflectance with a 4-parameter cumulative Gaussian. Fits were

obtained using a maximum likelihood method ( [36]) implemented by the psignifit toolbox in

Matlab (see http://bootstrap-software.org/psignifit/). Two parameters (!, ") determine the

7

0.2 0.3 0.4 0.50

0.5

1

Match Spot Reflectance

Prop

ortio

n M

atch

Jud

ged

Ligh

ter

Fig. 2. Example psychometric function for observer DBH. On each trial, the

reference spot reflectance was 0.32, the reference surround reflectance was 0.16,

the reference slant was 0 ", the match surround reflectance was 0.16, and match

slant was 0 ". The match spot reflectance was selected on each of 30 trials by

the staircase procedure. For visualization purposes, the 30 trials have been

divided into six bins of five trials each. In each bin, the reflectance values and

responses were averaged to get the x- and y- values for each plotted data point.

The horizontal bars show the variance of the match spot reflectances within

that bin. The black line represents the best fit cumulative gaussian to the data.

Black vertical line represents the extracted point of subjective equality (PSE),

or where the best fit curve reaches 50%.

8

shape of the cumulative Gaussian, and there is a floor parameter (#) and a ceiling parameter

($). The point of subjective equality (PSE) was defined as the reflectance at which observers

reported the match spot as lighter on 50% of trials. Each PSE was thus based on 30 forced-

choice trials. Figure 2 shows the data and fit for one reference spot in one match condition.

Pilot data indicated that within a subject, such PSEs collected on di!erent days were highly

consistent, and in fact were often identical within the reflectance resolution of our hidden

projector. Because of this consistency, we collected 2 PSEs per subject per condition.

3. Results

We measured the perceived lightness of small match spots across parametric changes of slant

and local surround reflectance. First, in Section 3.A we document that lightness constancy

was relatively high when match surround reflectance and slant were identical to reference

surround reflectance and slant. Then, in Sections 3.B and 3.C, we examine lightness con-

stancy under manipulations of match slant (Section 3.B), where local contrast is a valid cue

to reflectance, and under manipulations of match surround reflectance (Section 3.C), where

local contrast is not a valid cue. Finally, in Section 3.E, we examine interactions between

surround reflectance and slant.

3.A. Equal Slant and Surround

Figure 3 shows average PSEs for all six observers as a function of reference spot reflectance,

when the match surround reflectance was equal to the reference surround reflectance. In

this condition, the background squares on the left and the right have the same reflectance

(see Figure 1). Because the light source is to the left of the observer and angled across the

booth, the incident illumination at the left location (reference) is about twice the incident

illumination at the right location (match). If observers were lightness constant, then by

definition the reflectance of each PSE would be identical to the reflectance of the reference

spot. In other words:

Rmatch spot(PSE) = Rreference spot (1)

where R indicates reflectance values. The predictions of lightness constancy are shown as

the solid line in Figure 3. In general, observers exhibited good lightness constancy for this

condition, with some individual variation.

To understand the deviations from constancy, it is helpful to consider the pattern that

would be shown by an observer who matched the luminance of the spots rather than their

reflectance. This prediction is obtained by:

Lmatch spot(PSE) = Lreference spot (2)

9

0.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5

Reference Spot Reflectance

PSE

Refle

ctan

ce

SRA

CI = 0.750.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ce

DBH

CI = 0.970.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ce

IY

CI = 0.81

0.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ce

RTO

CI = 0.580.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ce

FP

CI = 0.830.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ceHB

CI = 0.73

Fig. 3. PSE as a function of reference spot reflectance for all six observers in

one condition (reflectance of match surround = 0.16, match slant = 0 "). Error

bars represent s.e.m. across sessions. Solid line represents PSE predictions

for both contrast matching and lightness constancy, which coincide for this

condition. Dashed line represents luminance matching predictions. The error-

based constancy index is reported in the lower right corner of each panel.

10

where L indicates reflected luminance. Because the illuminant intensity is less at the match

location than at the reference location, a much higher reflectance (PSE) is needed equate

luminance of the match spot and the reference spot. The predictions of luminance matching

are shown as the dashed lines in Figure 3.

When match surround reflectance and reference surround reflectance are the same, local

contrast is a valid cue to the reflectance of the match spot; in other words, the predictions of

local contrast matching are the same as predictions of lightness constancy (solid black line

in Figure 3. That is:

Rmatch spot(PSE) =

!Rreference spot

Rreference surround

"

! Rmatch surround. (3)

To quantify the degree of constancy, we calculated an error-based constancy index (after

[31]) for each subject by comparing the di!erence between the measured data point and the

predictions made from both lightness constancy and luminance matching, as follows:

CIerror =

#%2luminance#

%2luminance +

#%2constancy

(4)

Here each %2 is calculated as the sum of the squared error between each observed PSE

and the relevant prediction. The index can range from 0 to 1, where 1 represents perfect

lightness constancy (data along solid black line) and 0 represents luminance matching (data

along dashed black line, a failure of lightness constancy). Intuitively, the index characterizes

where the data fall with respect to the two di!erent predictions. In this condition, the CI

values of observers ranged from 0.58 to 0.97, as indicated in the individual plots.

3.B. Slant Manipulation

Observers were approximately lightness constant with respect to the illumination gradient

caused by the directional light source. To determine whether observers retained lightness

constancy across illumination changes mediated by other scene variables, we manipulated

match slant by rotating the stage on which the match card was mounted (see Figure 1).

Under this manipulation, we kept the match surround reflectance the same as the reference

surround reflectance. Rotating the stage changed the e!ective illumination incident at the

match location. However, since the reflectance of the surfaces did not change, local contrast

remained a valid cue to surface reflectance.

Figures 4 and 5 plot PSE as a function of reference spot reflectance when the match slant

was 10 "(Figure 4) and 20 " (Figure 5). If subjects were lightness constant, then the reflectance

of their PSEs should be una!ected by changing the match slant. This prediction is shown by

the solid line, which is unchanged across Figures 3, 4, and 5. Since rotating the stage changes

11

0.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ce

SRA

CI = 0.870.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ce

DBH

CI = 0.980.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ce

IY

CI = 0.81

0.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ce

RTO

CI = 0.670.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ce

FP

CI = 0.840.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ceHB

CI = 0.82



bars represent s.e.m. across sessions. Solid lines represents PSE predictions for

contrast matching and lightness constancy. Dashed lines represents luminance

matching predictions. The error-based constancy index is reported in the lower

right corner of each panel.

12

0.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ce

SRA

CI = 0.910.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ce

DBH

CI = 0.960.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ce

IY

CI = 0.87

0.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ce

RTO

CI = 0.810.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ce

FP

CI = 0.890.2 0.3 0.4 0.5

0.2

0.3

0.4

0.5


PSE

Refle

ctan

ceHB

CI = 0.88



bars represent s.e.m. across sessions. Solid lines represents PSE predictions for

contrast matching and lightness constancy. Dashed lines represents luminance

matching predictions. The error-based constancy index is reported in the lower

right corner of each panel.

13

the illumination incidental on the match, however, the luminance-matching predictions do

change. If perceived lightness followed luminance rather than surface reflectance, PSEs would

increase with slant (dashed lines in Figures 3, 4, and 5).

Observers’ PSEs were relatively constant across changes in slant; the data remain close

to the solid lines rather than deviating further towards luminance matching. The CI values

also reveal this constancy.

Together, Figures 3, 4 and 5 show two e!ects for each observer. The first (Figure 3) is

how much constancy the observer shows across a spatial illumination gradient. The second

(compare Figure 3 with Figures 4 and 5) is how much constancy the observer shows with

respect to a change in slant, once the e!ect of illumination gradient has been taken into

account. We can separate these two e!ect by normalizing the PSE at each match slant by

the PSE at 0 match slant. We then plotted PSE as a function of match slant for each reference

spot (Figure 6). This normalization preserves information about relative constancy across

changes in slant, but discards information about absolute constancy.

Observers exhibited very high degrees of lightness constancy across changes in slant, as

evidenced by the fact that for each reference spot (each di!erent color), the normalized

PSE stayed the same as slant changed (slopes of colored lines are near 0). If subjects were

perfectly lightness constant, PSE should not change with slant; that is, the slope of a line

through the points should be 0. However, if perceived lightness followed luminance rather

than surface reflectance, PSEs would increase with match slant (dashed black line). We

quantified the degree of relative constancy using the same constancy index as in Figures 3,

4, and 5, applied to the normalized PSE values. Constancy index values were close to 1 for

all six observers.

3.C. Reflectance Manipulation

When match slant was manipulated and local surround reflectance held fixed, perceived

lightness followed surface reflectance rather than luminance. However, local contrast under

this slant manipulation remained a valid cue to surface reflectance. Next, we examined

whether observers would show similar degrees of lightness constancy across a manipulation

where local contrast did not predict the reflectance of the match spot.

To do so, we held match slant fixed at 0 " and varied match surround reflectance. The

reference surround reflectance was always 0.16. We used match surround reflectances of

0.25, 0.34, 0.44 and 0.56.

To examine how perceived lightness of the match spot changed with match surround, we

again normalized PSEs by the PSEs in the condition where match surround reflectance and

match slant were the same as reference surround reflectance and reference slant (data in

Figure 3). The normalized PSEs as a function of match surround reflection are shown in

14

0 10 20

1

1.5

2

2.5SRA

Match Slant (degrees)

CI = 0.96

Norm

alize

d PS

E Re

flect

ance

0 10 20

1

1.5

2

2.5DBH


CI = 0.91

Norm

alize

d PS

E Re

flect

ance

0 10 20

1

1.5

2

2.5


Norm

alize

d PS

E Re

flect

ance

IY

CI = 0.86

0.180.200.220.260.32

0 10 20

1

1.5

2

2.5RTO


CI = 0.89

Norm

alize

d PS

E Re

flect

ance

0 10 20

1

1.5

2

2.5FP


CI = 0.87

Norm

alize

d PS

E Re

flect

ance

0 10 20

1

1.5

2

2.5HB


CI = 0.95No

rmal

ized

PSE

Refle

ctan

ce

Fig. 6. Normalized PSE as a function of match slant for all six observers. Each

color represents a di!erent reference spot reflectance. Reference surround was

equal to match surround (0.16). At each slant, the PSE was normalized by

the PSE at slant 0. Colored lines represent the best fit line through the data

points, when the line was constrained to go through the point (0, 1). Solid black

lines represent predictions of lightness constancy / contrast matching. Dashed

black line is predictions of luminance matching. Constancy index values are

calculated using normalized PSEs.

15

0.16 0.25 0.34 0.44

1

1.5

2

2.5SRA

Match Surround Reflectance

CI = 0.58

Norm

alize

d PS

E Re

flect

ance

0.16 0.25 0.34 0.44

1

1.5

2

2.5DBH


CI = 0.48

Norm

alize

d PS

E Re

flect

ance

0.16 0.25 0.34 0.44

1

1.5

2

2.5


Norm

alize

d PS

E Re

flect

ance

IY

CI = 0.54

0.180.200.220.260.32

0.16 0.25 0.34 0.44

1

1.5

2

2.5RTO


CI = 0.61

Norm

alize

d PS

E Re

flect

ance

0.16 0.25 0.34 0.44

1

1.5

2

2.5FP


CI = 0.29

Norm

alize

d PS

E Re

flect

ance

0.16 0.25 0.34 0.44

1

1.5

2

2.5HB


CI = 0.49No

rmal

ized

PSE

Refle

ctan

ce

Fig. 7. Normalized PSE as a function of match surround reflectance for all

six observers when match slant was equal to reference slant (0 "). Each color

represents a di!erent reference spot reflectance. At each match surround, PSE

was normalized by the PSE obtained in the condition where match surround

and reference surround were of equal reflectance (0.16). Colored lines represent

the best fit line through the data points, when the line was constrained to go

through the point (0, 1). Solid black lines represent predictions of lightness

constancy / luminance matching. Dashed black line shows predictions from

contrast. Constancy index values are calculated using normalized PSEs.

16

Figure 7. If observers were lightness constant across changes in match surround reflectance,

then their PSEs would not change and the data would fall along the dashed horizontal lines

in Figure 7. Changing the match surround reflectance a!ects the local contrast of the match

spot. If perceived lightness followed local contrast, then PSE would increase with increasing

match surround reflectance (angled dashed line in Figure 7).

The data show that local contrast a!ected perceived lightness. As match surround re-

flectance increased, PSEs also increased. However, although PSEs were a!ected by local

contrast, they were not completely determined by it. Normalized PSEs were significantly

lower than contrast matching predictions for all observers at each match surround reflectance

(p < 0.05, paired t-test) except one (observer fp, match surround 0.25). As in Figure 6, we cal-

culated an error-based constancy index, where data were compared to lightness constancy

predictions and contrast matching predictions. As with the previous index, 1 represents

complete lightness constancy. When match surround reflectance was varied, constancy index

values ranged from 0.29 (observer FP) to 0.61 (observer RTO).

3.D. Intermediate Discussion

Figure 8 summarizes the data reported above. When reference slant and surround reflectance

were the same as match slant and surround reflectance, observers were fairly lightness con-

stant across the illumination gradient, as seen by the high constancy index values in the

top left panel of Figure 8. Observers were also constant across changes in match slant (top

right panel, Figure 8), when local contrast was a valid cue to the surface reflectance of the

match spot. They were less constant across changes in match surround reflectance (bottom

left panel, Figure 8), when local contrast was not a valid cue.

An additional e!ect may be seen by closer examination of Figure 7: the e!ect of local

contrast depends on the reflectance of the reference spot. This is seen in Figure 7 by noting

the spread in the data for the separate reference reflectances. For each observer, the ma-

genta points (highest reflectance reference spots) are closer to lightness constancy (dashed

horizontal line) than are the red points (lowest reflectance test spot).

To document this e!ect, we compared two di!erent model fits to the data. We assumed

that the normalized PSEs in Figure 7 could be modeled as a linear function of match surround

reflectance constrained to go through the normalization point of (0.16, 1). We then compared

the model predictions made when PSEs for each reference spot were fit separately (colored

lines in Figure 7) to predictions made when all PSEs were modeled by one line (not shown).

We used the AIC (Akaike’s information criterion) [37] to compare the two models. This

criterion assigns scores to di!erent models, with a lower score meaning that a model is

preferred. Briefly, the likelihood of the data (L) given a maximum-likelihood fit to the the

model is calculated, and the model score decreases with increasing likelihood. The model is

17

SRA DBH IY RTO FP HB0

0.2

0.4

0.6

0.8

1

Reference Slant = 0 Match Slant = 0

Refe

renc

e Su

rroun

d =

0.16

M

atch

Sur

roun

d =

0.16


0.2

0.4

0.6

0.8

1

Reference Slant = 0 Match Slant = varied

Refe

renc

e Su

rroun

d =

0.16

M

atch

Sur

roun

d =

0.16


0.2

0.4

0.6

0.8

1

Reference Slant = 0 Match Slant = 0

Refe

renc

e Su

rroun

d =

0.16

M

atch

Sur

roun

d =

varie

d

Fig. 8. Constancy Index values for each subject. Each panel represents CI

values calculated in across a di!erent condition. Top left: match slant and

surround reflectance are the same as reference slant and surround reflectance.

Values reported in Figure 3. Top right: match surround reflectance and refer-

ence surround reflectance are the same, match slant is varied. Values reported

in Figure 6. Bottom left: match slant and reference slant are the same, match

surround reflectance is varied. Values reported in Figure 7.

18


50

100AI

C Di

ffere

nce

Slant Variation


50

100

AIC

Diffe

renc

e

Reflectance Variation

Fig. 9. Di!erence in the AIC score between a model in which normalized

PSEs for all reference spots are predicted by one line and a model in which

normalized PSEs are predicted by five lines, one for each reference spot. Each

bar is a di!erent subject. #AICs are shown for slant manipulation (left figure)

and match surround reflectance manipulation (right figure). Horizontal black

line indicates a #AIC of 10.

then penalized by the number of its free parameters (K).

AIC = #2ln(L(& | y) + 2K (5)

Two models can then be pitted against each other, with the di!erence between AIC scores

(#AIC) determining the extent to which one model is preferred over the other. #AIC values

of greater than 10 are taken to indicate that one model is significantly preferred ( [38]).

All subjects showed high #AICs in the direction of preferring the model that fit each ref-

erence reflectance separately. This means that the e!ect of manipulating the match surround

was dependent on the reflectance of the spot to which PSEs were being made (Figure 9).

For comparison, a similar analysis revealed that the e!ect of match slant was not dependent

on the reflectance of the reference spot (see low #AICs in the left panel of Figure 9 for all

but one observer). These #AICs are also consistent with the observation that, except for

observer IY, the best-fit lines for each reference spot shown in Figure 6 tend to lie on top of

each other.

3.E. Interactions between contrast, slant, and reflectance

Manipulations of slant and surround reflectance each change the luminance surrounding the

match spot. Made independently, these luminance changes had di!erent e!ects on perceived

lightness. That is, subjects were more lightness constant when luminance changes were in-

duced by manipulating slant than when luminance changes were induced by changing match

19

20 60 10020

60

100

140SRA

Match Surround Luminance (cd/m2)

PSE

Lum

inan

ce (c

d/m

2)

20 60 10020

60

100

140DBH


PSE

Lum

inan

ce (c

d/m

2)

20 60 10020

60

100

140RTO


PSE

Lum

inan

ce (c

d/m

2)

20 60 10020

60

100

140SRA


PSE

Lum

inan

ce (c

d/m

2)

20 60 10020

60

100

140DBH


PSE

Lum

inan

ce (c

d/m

2)

20 60 10020

60

100

140RTO


PSE

Lum

inan

ce (c

d/m

2)

Fig. 10. PSE as a function of match surround luminance for three ob-

servers for the lowest reflectance reference spot (top panels, reference spot

reflectance = 0.16) and the highest reflectance reference spot (top panel, refer-

ence spot reflectance = 0.32). Each color represents a di!erent slant (red = 0 ";

blue = 10 "; green = 20 ") and each symbol represents a di!erent match sur-

round reflectance (cross = 0.16; circle = 0.25; star = 0.34; diamond = 0.44;

square = 0.56). Black solid lines are predictions from contrast matching, and

colored solid lines are predictions for full lightness constancy at each slant.

Dashed lines represents maximum likelihood fits of the data to the modifed

Naka-Rushton function. Colored dashed lines fit data separately by slant, black

dashed line fit data from all slants simultaneously.

20

surround reflectance. To explore more completely the relationship between local contrast

and perceived lightness, we varied match slant and match surround reflectance parametri-

cally and investigated whether the e!ects of slant and surround reflectance could be modeled

with one function.

Figure 10 presents PSEs for 3 observers as a function of match surround luminance for all

14 match conditions, for two di!erent reference spots. Match slants are distinguished on the

plot by color, and match surround reflectances are distinguished by symbol shape. All data

points on a single panel were matches made to a single reference spot. Since both manipu-

lations (slant and reflectance) change the luminance of the local surround, we characterized

the PSE as a function of a single variable, the luminance of the match surround. Because

the analysis presented in Figure 9 suggested that the e!ects of match surround reflectance

were dependent on the reference spot reflectance, we considered separately PSEs made to

each reference spot.

Figure 10 confirms the conclusion we drew from the data in Figure 7: the full range of

data are not explainable as contrast matches (predictions shown as solid black line). It could

be, however, that the deviations from contrast matching are completely accounted for by the

photometric properties of the surround with no additional e!ect of slant. In this case, when

data are plotted as a function of surround luminance, they should fall on a common curve,

independent of slant. To test whether this was the case, we compared fits made to all data

simultaneously (black dashed lines, Figure 10) to fits made separately at each slant (colored

lines, Figure 10). This is the same type of model comparison used in the previous section

to test whether or not the e!ect of match surround reflectance on perceived lightness was

dependent on reference spot reflectance.

To fit the data in Figure 10, we used a the three-parameter Naka-Rushton function, as

follows:

Lmatch spot(PSE) = M(gLmatch surroundI)n

(gLmatch surround)n + 1. (6)

Parameters that best-fit the data were determined using parameter search in Matlab. Of

primary interest in modeling the data is whether a single function of luminance accurately

described PSEs in all 14 match conditions simultaneously, or whether the PSEs must be

separated into groups by slant in order to be well-fit. Our choice of parametric function was

somewhat arbitrary. The Naka-Rushton function has often been used because its parameters

are intuitively related to relevant psychophysical, physiological or physical variables; here its

use is dictated by its utility in describing the data.

To determine whether all PSEs for a particular subject and reference spot reflectance could

be well fit be a single function, we compared a model where PSEs for all 14 slant / surround

conditions were fit simultaneously (1 function / 3 parameters) to a model where PSEs were

21

SRA DBH IY RTO FP HB

0

50

100

Subject

AIC

Diffe

renc

e

0.180.200.220.260.32

Fig. 11. Di!erence in AIC score between the two models for each reference spot

reflectance (dark blue = 0.18, light blue = 0.20, green = 0.22, orange = 0.26,

red = 0.32) and each observer. Higher bars indicate that 9 parameters were

required to fit the data. Horizontal black bar represents #AIC of 10.

separated into three groups by slant (3 functions / 9 parameters).

For some observers and some reference spots, it was apparent that slant mattered. For

example, in the lower right panel of Figure 10, the three match surrounds outlined by the

black box had roughly equal luminance, though they di!ered in both slant and reflectance.

For this reference spot, the PSEs were clearly distinct, indicating that perceived lightness

was dependent on which combination of reflectance and slant determined a particular match

surround luminance. However, for the same observer in the same match condition, slant

played a less important role in perceived lightness when matches were made to a lower

reflectance reference spot (top right panel Figure 10). Though contrast alone could explain

PSEs made to this reference spot, the function of contrast that fit the data well (black,

dashed line) was not a simple ratio of local luminance values (solid black line). These are

two clear examples of when slant either mattered (bottom right panel) or did not (top right

panel). However, the degree to which slant mattered was not as obvious for other observers

and reference spot reflectances.

We compared AIC scores of model fits made to all data simultaneously with AIC scores

of model fits made to data at each slant separately. Figure 11 shows the #AIC between

the model in which slant mattered, and the model in which it did not. The height of the

bar represents the degree to which the model that takes into account slant was preferred.

#AICs greater than 10 are thought to indicate statistical preference. For each observer, slant

22

tended to play a more important role for higher reflectance reference spots. Although there

was variability between observers, we found statistical support for the full model at high

contrast reference spots for all observers except for one (FP).

To verify that the Naka-Rushton function characterized the data well, we also fit each PSE

with its own mean, so that PSEs were described by 14 parameters. We then calculated the

relevant AIC score. #AICs between the 14-parameter model (each point fit by its own mean)

and the 9- parameter model (PSEs separated by slant and fit with Naka-Rushton function)

were less than 10 for all but one reference spot for one subject (FP, data not shown). Small

#AICs mean that moving to 14 parameters from 9 parameters does not capture any more

variability in the data, indicating that 9-parameter Naka-Rushton model provided a good

description of the data.

4. Discussion

We measured perceived lightness across parametric changes in match slant and match sur-

round reflectance. These two manipulations both change the luminance of the match sur-

round, but a perfectly lightness constant system should treat those changes di!erently.

We found relatively high degrees of lightness constancy in the condition where match slant

and surround reflectance were equal to reference slant and surround reflectance (Figure 8,

top left panel). In this condition, there was an illumination gradient across the chamber.

The degree of lightness constancy we found (0.58 to 0.97) is similar to that reported in

previous studies of lightness and color constancy in both simulated and real-world scenes

( [23], [24], [39]).

Perceived lightness of a spot was highly constant across changes in match slant for all

subjects (Figure 6, Figure 8, top right panel) when match surround reflectance was fixed.

This finding stands in contrast to other studies that have shown much less lightness constancy

and higher inter-observer variability under slant manipulations ( [30], [31]). For example,

Ripamonti et. al. [31] reported that many observers were closer to luminance matching than

to constancy. Using the same constancy index, Ripamonti et al. reported CI values ranging

from 0.17 to 0.54, whereas we found CI values from 0.87 to 0.96 (Figure 8).

What can account for this large discrepancy in the results? A key di!erence between the

studies lies in what constituted the local surround of the match surface. In the Ripamonti et.

al. study ( [31]), the match surface was presented without an immediate coplanar surround,

so that the role of local contrast in these experiments is unclear. In the present study, the

match spot was immediately surrounded by another surface, as well as by the Mondrian

pattern (see Figure 1). In the special case where match surround reflectance was the same

as the reference surround reflectance, the local contrast between match spot and match

surround provided a valid cue to surface reflectance across variations in slant (Figure 6).

23

Several theories of lightness perception advocate the importance of local contrast cues in

determining perceived lightness, though these theories di!er in their emphasis and details

of how contrast cues are combined, especially for 3D scenes ( [19], [40], [41]). Because of

the di!erences in local contrast, the discrepancy between studies is not surprising. Note also

that in both studies, stimuli were presented within the same natural environment. Stimuli

were real, illuminated objects seen in depth in an experimental booth (see Figure 1). In

both studies, global context was su"ciently complex that changing slant should have been

unambiguously perceived. In fact, observers in the Ripamonti study made accurate judgments

about surface slant independent of their judgments about surface lightness. Therefore, the

higher lightness constancy in our study compared to the Ripamonti study is unlikely to have

resulted from a more accurate representation of slant itself.

Perceived lightness was less constant across changes in match surround reflectance (Fig-

ure 7) when match slant was fixed; in this circumstance local contrast was not a valid cue

to surface reflectance. Although observers exhibited less lightness constancy when match

surround reflectance was varied than when match slant was varied (range of CIs under re-

flectance manipulation: 0.29 - 0.61; range of CIs under slant manipulation: 0.87 - 0.96),

matches were still significantly di!erent than would be expected if local contrast were the

sole determinant of perceived lightness. Initially, we were surprised by this finding. Though

the match surround reflectance varied in di!erent conditions, the large Mondrian pattern

enclosing the immediate match surround was identical to the large Mondrian pattern enclos-

ing the reference surround. Mondrian patterns remained unchanged across all experimental

conditions and this provided an unambiguous cue that what was varied was match surround

reflectance, rather than match illumination. Thus, at first glance our findings seem at odds

with compelling visual illusions and quantitative studies which demonstrate that the role of

local contrast can be strongly mediated by global context or scene variable ( [19], [20], [42]).

We examined this apparent discrepancy by comparing the data from one such study ( [42]).

Using stimuli modeled after Adelson’s checkerboard illusion, ( [20]) Hillis and Brainard

demonstrated that even with equivalent local contrast, matches made in their ”paint” con-

ditions di!ered significantly from matches made in their ”shadowed” conditions. This is a

striking perceptual e!ect. However, when we compared the matches in Hillis and Brainard’s

”shadowed” condition to full lightness constancy predictions, we found substantial devia-

tions. Constancy index values in the Hillis and Brainard( [42]) study, computed with the

index we use here, were lower than those found in our experiments (Average CI = 0.13 for

”shadowed” conditions).

Our results show that local contrast has an important but not exclusive e!ect on perceived

lightness even in rich three dimensional scenes. This is consistent with the conclusion drawn

by Kraft and Brainard ( [24]) for color constancy. The e!ect of local contrast is weakened, but

24

not eliminated, when local contrast is not a valid cue to surface reflectance. In these cases,

perceived lightness is intermediate between contrast matching and lightness constancy. How-

ever, the weakening of local contrast is a large perceptual e!ect, as seen by the comparison of

the size of our e!ect with that of previously published data ( [42]). We caution that finding

a large perceptual e!ect in a study does not necessarily imply that the visual system has

achieved full lightness constancy.

When both match surround and match slant were parametrically varied, most of the

observers appeared to behave di!erently to changes in surround luminance caused by re-

flectance and changes in surround luminance caused by slant variation, and this tendency

was greater when the reference spots were of higher reflectance. However, there was a great

deal of inter-observer variability in the strength of this e!ect, and in its dependence in ref-

erence spot contrast (Figure 11). This inter-observer variability stands in contrast to high

intra-observer reliability, where PSEs for the same condition tested on separate days was

often identical within the resolution of our experimental apparatus. What can account for

this inter-observer variability? Though we put substantial e!ort into creating experimental

conditions and instructions that would elicit similar response strategies from all observers,

it remains possible that observers perceived surfaces similarly, but employed di!erent strate-

gies in making responses. Some studies have reported that in lightness tasks, observers can

make distinct judgments depending on instructions ( [23], [39]), though other studies have

found little dependence of responses on instructions, particularly in more realistic conditions

( [31]). Our pre-experiment induction procedure was specifically designed elicit the same

response strategy from all observers, and our instructions clearly directed observers towards

lightness matching. In addition, we used a 2AFC task rather than an adjustment task in an

attempt focus on perceptual rather than strategic processes.

If the observed inter-observer variability is perceptual rather than strategic, this would

be consistent with reports that have found high inter-observer variability in the e!ects of

geometry on various aspects visual perception ( [31], [43], [44]). One potential explanation is

that information about slant is encoded at a stage of visual processing that is more variable

from observer-to-observer than the mechanism that encodes local contrast.

In agreement with previous studies, we found that local contrast predicts perceived light-

ness better when it is a valid cue to surface reflectance than when it is not. However, our

findings suggest that even in a rich global context, perceived lightness never reaches full

constancy when there is a strong and opposing local contrast cue.

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